Signal integrity analysis of high-performance electronic systems requires knowledge and utilization of transmission-line parameters such as, e.g., per-unit-length resistance, inductance, conductance, and capacitance matrices (collectively “RLGC parameters”) and propagation constants. Transient simulation of transmission lines using their transmission-line parameters tends to be more accurate than direct simulation of network parameters, e.g., scattering (or S-) parameters, of the transmission lines. The scattering parameters typically come from measurements (made, for example, by a vector network analyzer), simulations from numerical three-dimensional electromagnetic solvers, or simulations from circuit solvers or closed-form expressions.
Procedures for the extraction of transmission-line parameters from tabulated network parameters are fairly well known, and typically involve solving the opposite problem—computing network parameters from transmission-line parameters—in reverse (see, e.g., W. R. Eisenstadt and Y. Eo, “S-parameter-based IC interconnect transmission line characterization,” IEEE Trans. on Components, Hybrids, and Manufacturing Technology, Vol. 15, No.4, pp. 483-490 (1992), the entire disclosure of which is incorporated by reference herein). Most extraction algorithms utilize a discontinuity-detection-based phase-unwrapping algorithm, which converts a sequence of cyclic phases to their noncyclic counterparts by adding integer multiples of 2π to each cyclic phase. The multiple is the total number of discontinuities in the sequence of cyclic phases observed between zero and a particular frequency (of a cyclic phase) that is more than π in amplitude. This algorithm is simple and computationally efficient, but it requires that the input cyclic phase constant not have any artificial discontinuities. In theory, the frequency-dependent phase constants of transmission-line systems are smooth functions of frequency (i.e., have no discontinuities). However, in practice, some artificial and unintentional discontinuities may be present in cyclic phase constants due to numerical artifacts in the extraction algorithm.
Further, the use of discontinuity-detection-based phase unwrapping requires that the tabulated network parameters are known for non-arbitrary frequencies. Specifically, the frequency step and starting frequency cannot be more than a particular data-dependent constant. This constant, a positive number, is inversely proportional to the propagation delay in the transmission lines. When these constraints are not met, the discontinuity-detection-based algorithm cannot be applied reliably. In particular, applying this algorithm to data not meeting the constraints results in the noncyclic phase constant computed from the unwrapping being arbitrary by an integer multiple of
            2      ⁢      π        l    ,where l is the length of the line; this arbitrariness results in incorrect values for per-unit-length inductance and capacitance parameters.
Other phase-unwrapping algorithms have been demonstrated in which, unlike the discontinuity-detection-based algorithm, the unwrapped phase at a particular frequency depends only on the wrapped phase at the same frequency, rather than also on the values of the unwrapped phases prior to the particular frequency (see, e.g., L. F. Knockaert, et al., “Recovering lossy multiconductor transmission-line parameter from impedance or scattering representations,” IEEE Trans. on Advanced Packaging, Vol. 25, No.2, pp. 200-205 (2002), hereafter the “Knockaert reference,” the entire disclosure of which is incorporated by reference herein). While such phase-unwrapping algorithms do not impose the above-described numerical challenges, they tend to be computationally inefficient, as their time complexities are exponential with the number of transmission lines. This time complexity increases with the electrical length of the lines. Further, no existing formulation properly handles singularities, resulting in nonphysical discontinuities in transmission-line parameters. Therefore, these alternative phase-unwrapping algorithms may be unsuitable for real-world problems, and there is a need for simulators and simulation methods for the extraction of transmission-line parameters from tabulated network parameters that utilize discontinuity-detection-based phase unwrapping and that are numerically reliable and computationally efficient.